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Complex Polynomial of nth degree

  1. Apr 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that if
    [tex]P(z)=a_0+a_1z+\cdots+a_nz^n[/tex]
    is a polynomial of degree [itex]n[/itex] where [itex]n\geq1[/itex] then there exists some positive number [itex]R[/itex] such that
    [tex]|P(z)|>\frac{|a_n||z|^n}{2}[/tex]
    for each value of [itex]z[/itex] such that [itex]|z|>R[/itex]

    2. Relevant equations
    Not sure.

    3. The attempt at a solution
    I've tried dividing through by the nth power of z. That way I can somehow incorporate the R value somehow but I'm not exactly sure where to go from here.

    Thanks!
     
  2. jcsd
  3. Apr 26, 2015 #2

    PeroK

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    Maybe you could start by showing that for large enough z,

    ##|z|^n > |a_0|##

    And, perhaps, rewrite the equation with everything but ##a_n z^n## on the LHS.

    Can you see, without doing any algebra, why it's true?
     
    Last edited: Apr 26, 2015
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